Formula

Length of a common tangent to two circles

Geometry • Circle • Tangents

Formula

L_{direct} = d^{2} - (r_{1} - r_{2})^{2}, L_{transverse} = d^{2} - (r_{1} + r_{2})^{2}

For two circles of radii

r_{1}

and

r_{2}

whose centres are

d

apart, the direct (external) common tangent has length

d^{2} - (r_{1} - r_{2})^{2}

and the transverse (internal) common tangent has length

d^{2} - (r_{1} + r_{2})^{2}

, the second existing only when

d \geq r_{1} + r_{2}

How to use it

1Both formulas come from one right triangle: drop a perpendicular from the smaller centre onto the radius of the larger, and the tangent length becomes the remaining leg.
2Direct tangents touch both circles on the same side of the line of centres and use the DIFFERENCE of the radii; transverse tangents cross between the circles and use the SUM.
3A transverse length of $0$ means the circles touch externally, and a direct length of $0$ means they touch internally.
4Look for a Pythagorean triple before reaching for a square root.

Worked example

Two circles of radii

8

cm and

3

cm have their centres

13

cm apart. Find the length of a transverse common tangent.

Step-by-step

1
$L = 1 3^{2} - (8 + 3)^{2} = 169 - 121$
Why: A transverse tangent uses the sum of the radii.
2
$L = 48 = 43$ cm

Answer

43

Watch out

Taking the sum where the difference belongs is the standard error, and it silently returns a shorter tangent that still looks reasonable; also, a negative quantity under either radical means that kind of common tangent does not exist, not that the arithmetic went wrong.

Builds on

The two tangents from an external point Pythagoras' theorem

Also known as

direct common tangent, transverse common tangent, external common tangent, internal common tangent, length of common tangent between two circles

Length of a common tangent to two circles

Formula

Worked example

Appears in past papers

2018 (I)

2017 (II)